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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Two dimensional elliptic equation with critical nonlinear growth


Authors: Takayoshi Ogawa and Takashi Suzuki
Journal: Trans. Amer. Math. Soc. 350 (1998), 4897-4918
MSC (1991): Primary 35J60, 35P30, 35J20
DOI: https://doi.org/10.1090/S0002-9947-98-02269-7
MathSciNet review: 1641254
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Abstract: We study the asymptotic behavior of solutions to a semilinear elliptic equation associated with the critical nonlinear growth in two dimensions. \begin{equation*}\tag {1.1} \begin {cases} -\Delta u= \lambda ue^{u^2}, u>0 & \text {in $\Omega $}, \\ u=0 & \text {on $\partial \Omega $}, \end{cases} \end{equation*} where $\Omega$ is a unit disk in $\mathbb {R}^2$ and $\lambda$ denotes a positive parameter. We show that for a radially symmetric solution of (1.1) satisfies \begin{equation*} \int _{D}\left \vert \nabla u\right \vert ^{2}dx\rightarrow 4\pi ,\quad \lambda \searrow 0. \end{equation*} Moreover, by using the Pohozaev identity to the rescaled equation, we show that for any finite energy radially symmetric solutions to (1.1), there is a rescaled asymptotics such as \begin{equation*} u_m^2(\gamma _m x)-u_m^2 (\gamma _m)\to 2\log \frac {2}{1+|x|^2} \quad \text {as }\lambda _m\searrow 0 \end{equation*} locally uniformly in $x\in \mathbb R^2$. We also show some extensions of the above results for general two dimensional domains.


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Additional Information

Takayoshi Ogawa
Affiliation: Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California 93106
Address at time of publication: Graduate School of Mathematics, Kyushu University 36, Fukuoka, 812-8581, Japan
MR Author ID: 289654
Email: ogawa@math.kyushu-u.ac.jp

Takashi Suzuki
Affiliation: Department of Mathematics, Osaka University, Toyonaka, Osaka 560, Japan
MR Author ID: 199324
Email: takashi@math.sci.osaka-u.ac.jp

Received by editor(s): January 29, 1996
Additional Notes: The first author is on long-term leave from the Graduate School of Polymathematics, Nagoya University, Nagoya 464-01 Japan.
Article copyright: © Copyright 1998 American Mathematical Society